Correct Answer - b
We have , `I= int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx`
`rArr I= int(sec^(2)x)/(b^(2)+a^(2)tan^(2)x)dx`
` rArr I=(1)/(a)int(1)/(b^(2)+(atanx)^(2))d(atanx)`
`=(1)/(ab) tan^(-1)((a)/(b)tanx)+C`
`:. ab = 12 and (a)/(b) = 3 rArr a^(2)= 36 rArr a = +-6`
`:. ab = 12 rArr b =+-2`.
Thus , we have
`a sin x +b cos x = +- (6 sin x+2cosx)` We know that
`-sqrt(a^(2)+b^(2))le a sin x +b cos x le sqrt(a^(2)+b^(2))` for all x
` :. - sqrt(40)le +- 6 sin x +- 2 cos x le sqrt(40)` for allx.
